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$\mathcal {W}$-constraints for the total descendant potential of a simple singularity
Simple, or Kleinian, singularities are classified by Dynkin diagrams of type 
$ADE$. Let 
$\mathfrak {g}$ be the corresponding finite-dimensional Lie algebra, and 
$W$ its Weyl group. The set of 
$\mathfrak {g}$-invariants in the basic representation of the affine Kac–Moody algebra 
$\hat {\mathfrak {g}}$ is known as a 
$\mathcal {W}$-algebra and is a subalgebra of the Heisenberg vertex algebra 
$\mathcal {F}$. Using period integrals, we construct an analytic continuation of the twisted representation of 
$\mathcal {F}$. Our construction yields a global object, which may be called a 
$W$-twisted representation of 
$\mathcal {F}$. Our main result is that the total descendant potential of the singularity, introduced by Givental, is a highest-weight vector for the 
$\mathcal {W}$-algebra.
